Question

The annual spending by tourists in a resort city is \(\$ 100\) million. Approximately \(75 \%\) of that revenue is again spent in the resort city, and of that amount approximately \(75 \%\) is again spent in the same city, and so on. Write the geometric series that gives the total amount of spending generated by the \(\$ 100\) million and find the sum of the series.

Short answer

The total amount of spending generated by the $100 million is $400 million.

Step-by-step solution

Identify the first term and common ratio

The first term is given in the problem as the initial spending by tourists, which is $100 million. Each subsequent term represents the amount spent in the city from the revenue of the previous term. Hence, the common ratio is 0.75.

Write the geometric series

The geometric series is written as follows: S = \(100 + 100(0.75) + 100(0.75^2) + 100(0.75^3) + ... \)

Find the sum of the series

The sum (S) of an infinite geometric series can be calculated using the formula S = \(a / (1-r)\), where a is the first term and r is the common ratio. Substituting the given values, we get S = \(100 / (1 - 0.75) = 400\) (in millions). Hence, the total amount of spending generated by the $100 million is $400 million.